PSI - Issue 19

H. Heydarinouri et al. / Procedia Structural Integrity 19 (2019) 482–493 H. Heydarinouri et al. / Structural Integrity Procedia 00 (2019) 000 – 000

489

8

doesn’t usually change the stress range significantly (Ghafoori and Motavalli, 2015, Ghafoori and Motavalli, 2016, Ghafoori et al., 2015c), while it decreases the stress ratio as shown in Fig. 4-b and c. In other words, the mean stress m  is reduced to the lower value of * m  as depicted in this figure. The goal is to obtain the minimum prestressing force P , required for a system with the eccentricity of e (see Fig. 4-a), according to the proposed criterion. As shown in Fig. 3, before strengthening, the stress ratio is R . Application of prestressing force reduces the stress ratio to R *, with the stress range remaining constant (path (1) in Fig. 3). Us ing Eq. (10), the reduced stress ratio R *, corresponding to which is on the curve of the proposed criterion (see Fig. 8), is obtained by Eq. (19): * 1 ut R     (19)

S k



 

f

* max  after application of prestressing force (see Fig. 4-c), according to Eq. (2) and

The maximum stress level

Eq. (19), is obtained by the following equation: * max 

S k

ut    

(20)

f

Cross-section analysis after application of prestressing force of P implies that: * max max net net net 1 ( ) M e P S S A     Where max M is the maximum bending moment due to the external load, and, net S and net A are the section modulus and the net section area, respectively. Considering the fact that the stress range   , and the stress ratio before strengthening, R , have the following relationship with the first term of the right side of Eq. (21): (21)

M S

R   

max



(22)

1

net

Therefore, using Eq. (20), (21), (22) and (16), the prestressing force P is obtained by Eq. (23):

2 1

1

R

e



(

) / (

)

P

 



 



(23)

R

net S A



net

Using the value of 144   (see Table 1), for the design purpose, the required prestressing force is only depend ent on the stress ratio of the applied load as well as the section properties of the riveted member. For the beam shown in Fig. 4-a, which is subjected to a constant amplitude loading with R =0.1 (see Fig. 3), the required prestress ing force is: 723 kN P  , using Eq. (23). The important point is that according to the Eurocode criterion, with such stress range and stress ratio, it is nearly impossible to prevent fatigue crack in this riveted member, shown in Fig. 3, since it only accounts for the effect of R ratio for R < 0. 4.2. Non-prestressed retrofitting systems When non-prestressed strengthening systems are used, the stiffness of the structural members increases, resulting in reduction of the stress ranges. Thus, as shown in Fig. 4-d, the nominal stress range decreases. If the dead load of the riveted member is negligible compared to the superimposed live loads, the stress ratio can be assumed to remain constant. Therefore, the points which are beyond the fatigue limit curves, shown in Fig. 3, follow the path (2), as a result of the strengthening. In order to determine the additional stiffness required to bring the point from above the CAFL to below it, it is essential to know which failure criterion is selected; i.e. either the curve proposed in this study or that presented in Eurocode EN 1993-1-9 (2005) (see Fig. 3). In order to be inside of the zone of infinite fatigue life based on the proposed criterion, the maximum stress range in the beam has to be reduced from   to at least *   (see Fig. 3). Using Eq. (17), along with the fact that * * net net S S        , the minimum section modulus required for reducing the stress range from   to *   , is ob tained as: